The solution to coin distribution offered here is different. I suggest that there needs to be two types of coin: coin that acts like money and coin that acts like shares in the system's seigniorage. For short, we'll just call these coins and shares. Coins and shares are identical in all respects (transaction verification works via ECDSA signature, etc) except for the process that regulates their respective supply.
Coins are the object of stabilisation, and _i of coin is distributed to the holders of shares. When coin supply needs to increase, coin base is distributed to shareholders in exchange for a certain percentage of shares, which are destroyed (coin supply increases, share supply decreases). When coin supply needs to decrease, share base is distributed to coin holders in exchange for a certain percentage of coin, which are destroyed (coin supply decreases, share supply increases).
The mechanism of these shares-for-coin and coin-for-shares swaps is a voluntary one, a decentralized auction the rules of which are written into the protocol. The quantity of coins to create or destroy is defined via the _i process in equations 3 and 4, and there is an auction at the end of every rebase period. When _i is positive (new coins need to be created) a coin auction for _i coins begins at the block or ledger set defining the start of rebase period i+1.
Any holder of shares can bid for coins by signing and broadcasting a special TX describing the quantity of shares he is willing to trade for coin, and the minimum coins/shares price that he will accept. Winning bids are filled at whatever coins/shares price Ps that clears the quantity to be sold. So _i new coins are distributed to the winning bidders and _i=Ps shares are destroyed.
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